CR circuit: harmonic response

Control theory

In this article, we will deal with a simple high pass filter, made using a capacitor and a resistor, connected in series. We already saw something similar in the article about a RC circuit, but this time the output voltage will be measured on the resistor instead of the capacitor. Obviously the circuit will behave in a completely different way. For the sake of simplicity, the calculations will be carried out in the Laplace domain.

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Using the voltage divider formula, we can write:

V_u=V_i\frac{Z_R}{Z_R+Z_C}=
=V_i\frac{R}{R+\frac{1}{sC}}=V_i\frac{R}{\frac{RCs+1}{sC}}

Simplifying the fraction and dividing by V_i, we can write:

G(s)=\frac{V_u}{V_i}=\frac{RCs}{1+RCs}

We can now identify the parameters in the transfer function. From the formula above we can easily see that there is a binomial factor in the denominator. In the numerator we can identify a zero in the origin and two constants, R and C that will be part of the static gain K. As usual these values refer to the Bode form formula, reported below.

K\frac{(1+T_1s)+...(1+\frac{2\zeta_1s}{\rho_{n1}}+\frac{s^2}{\rho_{n1}^2})+...}{s^n(1+\tau_1s)+...(1+\frac{2\xi_1s}{\omega_{n1}}+\frac{s^2}{\omega_{n1}^2})+...}

As a result we will choose the folowing parameters.

Parameter Value
K RC
n -1
\tau_1 Time constant in the denumerator. There is just one: RC